4.7 Inversion of a Square Matrix

Matrix Inverse

A square matrix $A\in {\mathbb{R}}^{n\times n}$ is said to be invertible if it is nonsingular (i.e. $\det (A)\ne 0$), such that there exists a matrix $A^{-1}\in {\mathbb{R}}^{n\times n}$ for which
$A{A}^{-1}=A^{-1}A=I_n.$
If only one of the above conditions hold, we say that matrix $A$ is left-invertible (such that $A^{-1}A=I$), or right-invertible (such that $A{A}^{-1}=I$).

We can find the inverse of a matrix $A\in {\mathbb{R}}^{2\times 2}$ by dividing its transpose by its determinant so that
$A^{-1}=\frac{1}{\det (A)}\left[\begin{array}{cc}d& -c\\ -b& a\end{array}\right].$

Furthermore, if two square matrices $A,B\in {\mathbb{R}}^{n\times n}$ are invertible, it holds that $(AB{)}^{-1}=A^{-1}{B}^{-1}$.

Intuition

The matrix inversion formula just says that given a linear transformation of the n-space $A$, its inverse will reverse that transformation so when $A{A}^{-1}=A^{-1}A=I$ is applied, there is no net change for any vector in the space by yielding theidentity matrix $I_n$.